Given y 2xy y1 1 Find y12 using the Improved Euler method

Given: y\' = 2xy,     y(1) = 1

Find y(1.2) using the Improved Euler method with a step size of h=0.1. Round the answer to four decimal place.

Differential Equations Problem, please show work so it is easy to follow. Will rate, thanks!

Solution

The derivative term in the first order ivp

y\' = f(x, y) , y(x₀) = y₀

is approximated by making use of Taylor series approximation of the dependent variabley(x) at the point x_i+1. That is

y(x_i+1) = y(x_i+ Dx) = y(x_i) + Dxy\'(x_i) + (Dx² / 2)y\'\'(x_i) + . . .
            = y(x_i) + Dxf(x_i, y_i) + (Dx² / 2)y\'\'(x_i) + . . .

(... y\'(x_i) = f(x_i, y_i))

if the infinite series is truncated from the term Dx² onwards, then

y(x_i+1) = y(x_i) + Dx y\'(x_i) (or)
y_i+1 = y_i + Dx f_i    for all i

That is,
for i = 0,      y₁ = y₀ + Dx f₀
     i = 1,      y₂ = y₁ + Dx f₁
     !
     i = n-1,   y_n = y_n-1 + Dx f_n-1

Since y₀ and hence f₀ are known (from initial condition) in the equation corresponding to i = 0, all the terms on the r.h.s are known. So y₁ that is, y at x₁ is calculated easily from this equation. Similarly once y₁ is known, r.h.s of the equation corresponding to  i = 1 is also known so y₂ can be computed. As we proceed in the same way until i = n-1, y_n can be obtained. This is an explicit method because in any equation there is only one unknown which can be separated to the left side of the equation.